Page 84 - Physical Chemistry

lev38627_ch02.qxd 2/29/08 3:11 PM Page 65

65
2
However, in evaluating C (T) dT, we cannot take C outside the integral, unless we Section 2.10
1 P P
know that it is constant in the temperature range from T to T . State Functions and Line Integrals
1 2
EXAMPLE 2.6 Calculation of H

C of a certain substance in the temperature range 250 to 500 K at 1 bar pres-
P,m
sure is given by C b kT, where b and k are certain known constants. If n
P,m
moles of this substance is heated from T to T at 1 bar (where T and T are in
1 2 1 2
the range 250 to 500 K), find the expression for H.
Since P is constant for the heating, we use (2.79) to get
P 2 T 2 1 2 T 2
¢H q nC P,m dT n 1b kT2 dT n1bT kT 2`
2
1 T 1 T 1
1
2
2
¢H n3b1T T 2 k1T T 24
1
1
2
2
2
Exercise
1/2
Find the H expression when n moles of a substance with C r sT ,
P,m
where r and s are constants, is heated at constant pressure from T to T .
1
2
2
[Answer: nr(T T ) ns(T 3/2 T 3/2 ).]
2 1 3 2 1
2.10 STATE FUNCTIONS AND LINE INTEGRALS
We now discuss ways to test whether some quantity is a state function. Let the system
go from state 1 to state 2 by some process. We subdivide the process into infinitesimal
steps. Let db be some infinitesimal quantity associated with each infinitesimal step.
For example, db might be the infinitesimal amount of heat that flows into the system
in an infinitesimal step (db dq), or it might be the infinitesimal change in system
pressure (db dP), or it might be the infinitesimal heat flow divided by the system’s
temperature (db dq/T), etc. To determine whether db is the differential of a state
2
function, we consider the line integral db, where the L indicates that the integral’s
L 1
value depends in general on the process (path) used to go from state 1 to state 2.
2
The line integral db equals the sum of the infinitesimal quantities db for the
L 1
infinitesimal steps into which we have divided the process. If b is a state function, then
the sum of the infinitesimal changes in b is equal to the overall change b b b
2 1
in b from the initial state to the final state. For example, if b is the temperature, then
2
2
dT T T T ; similarly, dU U U . We have
L 1 2 1 L 1 2 1
2 db b b if b is a state function (2.82)
2
1
1
L
Since b b is independent of the path used to go from state 1 to state 2 and depends
2
1
2
only on the initial and final states 1 and 2, the value of the line integral db is inde-
L 1
pendent of the path when b is a state function.
Suppose b is not a state function. For example, let db dq, the infinitesimal heat
flowing into a system. The sum of the infinitesimal amounts of heat is equal to
the total heat q flowing into the system in the process of going from state 1 to state 2;
2
2
we have dq q; similarly, dw w, where w is the work in the process. We
L 1
L 1
have seen that q and w are not state functions but depend on the path from state 1 to